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MySQL Replication Issue

I have mysql master database and 10 slave already running replication. I want to add a new mysql database server and want to enable replication on it. The thing is that i can't take dump of database from master because i don't want to block updates on MASTER & because of load issue. My database size is around 20GB which i want to replicate. I took database dump from one of the slaves and i followed the following steps.

mysql-slave> stop slave;

mysql-slave> show slave status;

Master_Log_File: mysql-bin.00010
Master_Log_Pos: 0020

mysql-slave> mysqldump database > database.sql

once it took the complete backup of database i started the slave again

mysql-slave> start slave;

Now in the mean time master contuniously updating the database. and it's changed it's log file and position.

mysql-master> show master status;

Master_Log_File: mysql-bin.00020
Master_Log_Pos: 0040

After that i switched to new mysql server where i want to enable replication.

* What i believe i am getting 'Duplicate Entry Error' because the time when i am taking dump because DB is almost 20 GB and it took 1/2 an hour for that and in the mean time it's continuously updating master and it's missed the log file between that time.

I don't know i am right or not. Please help me out and let me know what should be the problem ? Thanks in advance